All Your Bases Belong to Me: Exploring Exponent Rules

Welcome to Professor Baker's Math Class! Today, we're diving into the fascinating world of exponents. Understanding exponent rules is crucial for simplifying expressions and solving more complex math problems. Let's get started and make sure all your bases (and exponents!) belong to you!

What is a Power?

In mathematics, a power is an exponential expression that indicates the number of factors involved in multiplication. It consists of two main parts:

  • Base: The number that gets multiplied.
  • Exponent: The number that tells you how many times to multiply the base by itself.

We can represent a power as $b^e = n$, where:

  • $b$ is the base
  • $e$ is the exponent
  • $n$ is the resulting number

For example, in $3^4 = 81$, 3 is the base, 4 is the exponent, and 81 is the result of multiplying 3 by itself four times: $3 \cdot 3 \cdot 3 \cdot 3 = 81$.

Exponent Rules

Here are the key exponent rules you need to know:

  1. Multiplying Powers with the Same Base: When multiplying powers with the same base, add the exponents.

    2. Example: $a^m \cdot a^n = a^{m+n}$. So, $a^3 \cdot a^2 = a^{3+2} = a^5$

  2. Dividing Powers with the Same Base: When dividing powers with the same base, subtract the exponents.

    3. Example: $\frac{a^m}{a^n} = a^{m-n}$. So, $\frac{a^5}{a^3} = a^{5-3} = a^2$

  3. Raising a Power to Another Power: When raising a power to another power, multiply the exponents.

    4. Example: $(a^m)^n = a^{m \cdot n}$. So, $(a^4)^2 = a^{4 \cdot 2} = a^8$

  4. Powers with a Negative Exponent: A power with a negative exponent can be written as a fraction with a positive exponent.

    5. Example: $a^{-n} = \frac{1}{a^n}$. So, $a^{-5} = \frac{1}{a^5}$. Conversely, $\frac{1}{a^n} = a^{-n}$.

  5. A Power with an Exponent of One: Any number raised to the power of 1 is equal to the number itself.

    6. Example: $a^1 = a$

  6. A Power with an Exponent of Zero: Any non-zero number raised to the power of 0 is equal to 1.

    7. Example: $a^0 = 1$ (where $a \neq 0$)

  7. Product to a Power Rule: When a product is raised to a power, each factor is raised to that power.

    8. Example: $(ab)^n = a^n b^n$. So, $(xy)^2 = x^2 y^2$

  8. Quotient to a Power Rule: When a quotient is raised to a power, both the numerator and denominator are raised to that power.

    9. Example: $(\frac{a}{b})^n = \frac{a^n}{b^n}$. So, $(\frac{x}{y})^3 = \frac{x^3}{y^3}$

Remember, practice makes perfect! Work through the class worksheet and homework to solidify your understanding of these rules.

Don't forget to ponder this: What happens when exponents are fractions? Do the rules hold, or do they morph? Share your thoughts and examples in the discussion below!

Good luck, and happy calculating!

Resources:

Here are the resources for today's lesson:

  • Notes on Exponent Rules - Notes 1 and Notes 2
  • Class Worksheet
  • Homework - Practice 6-1 Worksheet